# Questions tagged [complex-manifolds]

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### Compact connected complex/holomorphic manifold that embeds in $\mathbb{C}^n$

If $i : M \to \mathbb{C}^n$ is a holomorphic map, then each coordinate function on $\mathbb{C}^n$ restricts to a global holomorphic function on the image. In particular, there is no holomorphic ...
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### Analytic variety induced by an irreducible polynomial

I am reading Principles of Algebraic Geometry by Griffith and Harris. Here the authors define an analytic variety in a domain as follows : A subset $V$ of an open set $U \subset \mathbb{C}^n$ is an ...
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### Integrable almost complex structure on a torus

Consider the torus $T^2 = S^1 \times S^1$ and let $(x,y)$ be the canonical coordinates $(0<x<2\pi, 0<y<2\pi)$ on $T^2$. The corresponding coordinate vector fields define global fields ...
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### Holomorphic maps of complex manifolds preserve the bidegree of a complex differential form

For a holomorphic map $f : M \to N$ between complex manifolds show that if $\omega$ is a form of type $(p,q)$ on $N$, then $f^*\omega$ is a form of type $(p,q)$ on $M$. I am wondering if the ...
1 vote
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### Proof of the fact that $\mathfrak{g}\otimes C^{\infty}(S^1) \cong C^{\infty}(S^1;\mathfrak{g})$

I encountered the following fact: let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, and consider the tensor product $\mathfrak{g}\otimes_{\mathbb{C}} C^{\infty}(S^1)$ (where I guess we mean maps ...
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### Proof of Spherical metric on Riemann Sphere

While studying stereographic projection of extended complex on unit sphere $S$ in $\mathbb{R^3}$ we get two metrics one is chordal metric and second one is spherical metric. The spherical metric $d_s$ ...
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### Definition of meromorphic function between complex manifolds

Ususally we only consider meromorphic function from a complex manifold $X$ to $\mathbb{C}$: Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is sheaf of holomorphic functions. We ...
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### What is the definition of sheaf of meromorphic differential form on a complex manifold？

Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is the sheaf of $\mathbb{C}$-valued holomorphic functions on $X$. Let $T_X^\vee$ be the cotangent bundle over $X$ with the ...
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### fibres varying non-holomorphically

I'm currently reading Claire Voisin's book Hodge theory and complex manifolds. Here is an extract from it: in the last paragraph she writes that we cannot chose the trivialisation $T$ to be ...
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### $f:M \rightarrow N$ holomorphic between equidimensional complex manifolds is surjective if $|J(f)| \not\equiv 0$

In the book "Principles of algebraic geometry" by Griffiths and Harris (PG. 237) there is a proof of the following statement: "Let $f:M \rightarrow N$ be a holomorphic map between two ...
1 vote
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### Partial derivatives chart-dependent?

Let $X$ be a Riemann surface. Let $Y\subset X$ be open. The following definitions are taken from page 60 of Forster's Riemann Surfaces. We call a function $f\colon Y \rightarrow \mathbb{C}$ (...
1 vote
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### Singular irreducible affine plane curve is never a Riemann surface?

Let $f\in \mathbb{C}[x,z]$ be a bivariate irreducible polynomial over the complex numbers. In case that $f$ is non-singular, one can endow the locus (zero set) of $f$ (considered with subspace ...
1 vote
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### Multiplicity of a non-constant holomorphic map

In Miranda's Algebraic Curves and Riemann Surfaces he defines the multiplicity of a non-constant holomorphic map $F\colon X \rightarrow Y$ between Riemann surfaces as the unique integer $m$ such that ...
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### Source for nomenclature "metric tensor" on Hermitian manifold

I am a wikipedia editor and am looking for reliable sources to cite for the term metric tensor in the context of Hilbert spaces or Hermitian manifolds.
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### Expressing "the current $T$ gives no mass to the subset $E$" in terms of differential forms on the Complex Projective Plane $\mathbb{CP}^2$

Let $(\mathbb{CP}^2, \omega)$ be the Complex Projective Plane, where $\omega$ is a Hermitian Metric (or, the Kähler Form). Let $D^{(1,1)}(\mathbb{CP}^2)$ be the space of $(1,1)$-differential forms on ...
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### Holomorphic forms are closed on compact manifold $X$ if $\dim(X)=2$.

Let $X$ be a compact complex manifold and $\dim(X)=2$, $\eta$ is a holomorphic form on $X$. Prove that d$\eta=0$. I know when $X$ is a compact complex Kähler manifold, holomorphic forms are closed. In ...
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### Chern class of a hypersurface in $\mathbb{C}P^3$

Let $X=\{[z_0,z_1,z_2,z_3]\ \big{|}\ [z_0,z_1,z_2,z_3]\in\mathbb{C}P^3,z_0^4+z_1^4+z_2^4+z_3^4=0\}$. $c_1(X)$ is the first Chern class of $X$. Prove that $c_1(X)=0$. $\textbf{My try}$: It's easy to ...
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### affine Gauss map of Hypersurface Manifold finite

Let $M \subset \mathbb{C}^n$ be an algebraic $n-1$-dimensional manifold given as vanishing hypersurface of a polynomial $F \in \mathbb{C}[x_1,..., x_n]$ of degree $d \ge 2$. The smoothness of $M$ can ...
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### Is a finite covering of a $\partial\bar{\partial}$-manifold still $\partial\bar{\partial}$-manifold?

A compact complex manifold is called a $\partial\bar{\partial}$-manifold if for every pair $p,q\in \mathbb{N}$, every smooth $d$-closed $(p,q)$-form $\eta$ on $X$, $\eta$ is $d$-exact iff $\partial$-...
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1 vote
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### Regarding the Defintion of Hirzebruch Surfaces $\mathbb{F}_n$

We have $\mathbb{F}_0= \mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ and, for $n \geqslant 1$, the $n-$th Hirzebruch Surfaces $\mathbb{F}_n$ is defined as a $\mathbb{C}\mathbb{P}^1$-bundle ...
1 vote
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### Finite dimensional space of vector fields

Consider a tangent bundle $TM$ and the space of sections $\mathfrak{X}(X)$. In general, for a $C^\infty$ structure, this space is an infinite dimensional vector space over $\mathbb{R}$. I'm trying to ...
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### Let $z=a e^t$ then $dz=ae^tdt$, How to define $d \overline{z}=?$

Let $z$ and $t$ be two complex variables such that $z=a e^t$ where $a$ is a (real or complex) constant. Thus, the differential form $dz$ is nothing but $ae^tdt$, i.e., $$dz=ae^tdt.$$ The question is: ...
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1 vote
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### Question about complexified tangent bundle

In his book, Daniel Huybrechts define the complexified tangent bundle as: $T_{\mathbb C}U:= TU \otimes \mathbb C$. But I don't understand this tensor product, I understand what $T_xU \otimes \mathbb C$...
1 vote
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### Pulling-Back a Current by a Holomorphic Proper Non-Submersion Function Between Two Compact Complex Surfaces

Let $M,N$ be two Compact Complex Surfaces (compact complex manifolds of complex dimension $2$). Let $A$ be a (non-empty) subset of $M$ (not necessarily a sub-manifold). Let $f: M \longrightarrow N$ be ...
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### How to Push-Forward Differential Forms on a Complex Surface to a Complex Surface by a Holomorphic Function

Let $M$ and $N$ be two Compact Complex Surfaces (i.e., $M$ and $N$ are Compact Complex Manifolds of complex dimension Two). Let $A$ be a (non-empty) subset of the complex surface $M$ (not necessarily ...
1 vote
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### Under what conditions can we push-forward the differential forms on a manifold $M$ to differential forms on a manifold $N$ by a smooth map $f$?

Let $M,N$ be two Complex Manifolds of the same (complex) dimension. Let $f: M \longrightarrow N$ be a Smooth Map. It is well-known that differential forms on the manifold $N$ can always be pulled-back ...
Let $M,N$ be two complex manifolds of the same (complex) dimension. Let $\omega$ be a differential form on the manifold $N$ whose support denoted by $S$. Let $f: M \longrightarrow N$ be a smooth map. ...